Whether
it's your car crossing the finish line at 180 miles per hour, or a bullet
tearing through the chest of your best friend, all games make use of
collision detection for object interaction. This article describes some
simple intersection tests for the most useful shapes: spheres and boxes.

Sweep
Tests for Moving Objects

A
common approach to collision detection is to simply test for whether
two objects are overlapping at the end of each frame. The problem with
this method is that quickly moving objects can pass through each other
without detection. To avoid this problem, their trajectories can be
subdivided and the objects checked for overlap at each point; however,
this gets expensive if either object experienced a large displacement.
On the other hand, a sweep test can efficiently determine a lower and
upper bound for the time of overlap, which can then be used as more
optimal starting points for the subdivision algorithm.

A
Sphere-Plane Sweep Test

Figure
1 shows an example of a quickly moving sphere passing through a plane.
It can be seen that C0 is on the positive side of
the plane and C1 is on its negative side.

Figure
1. A sphere passes through a plane.

In
general, if a sphere penetrated a plane at some point during the frame,
then d0>r and d1<r, where
r is the radius of the sphere and d0 and d1
are the signed distances from the plane to C0 and
C1, respectively. The signed distance from a point
C to a plane can be calculated with the formula

More
efficiently, we can store the plane in the form {n, D},
where

The
distance d is then calculated

The
trajectory from C0 to C1 can be
parameterized with a variable u, which may be thought of as normalized
time, since its value is 0 at C0 and 1 at C1.
The normalized time at which the sphere first intersects the plane is
given by

The
center of the sphere at this time can then be interpolated with an affine
combination of C0 and C1

This
formula interpolates Ci correctly as long as d0
is not equal to d1 (which is the case if displacement
has occurred), even when r = 0 (the case of a line segment).
If desired, the parameter u can also be used to linearly interpolate
the orientation of an object at this point.

In
this example, it was assumed that the sphere approached the plane from
the positive side and that the sphere was not already penetrating the
plane at C0.. In the case that there could have been
penetration on the previous frame, the condition |d0|<=r
should also be checked. Listing 1 gives an implementation of this sphere-plane
sweep test.

Listing
1. A sphere-plane sweep test.

#include "vector.h"

class PLANE

{

public:

VECTOR N;
//unit normal

SCALAR D;
//distance from the plane to the origin
from a
//normal and a point